Abstract Herein we establish a relation between quantum irreversibility and the chaotic semi-classical solutions for a spin-boson Hamiltonian system. We obtain quantum averages by numerically integrating the appropriate Liouville-Von Neumann equations of motion and find these averages to be less erratic than the corresponding chaotic semi-classical trajectories. However, the quantum averages are shown to be dissipative as measured by the entropy of the spin subsystem and to suppress the phenomenon of "revivals".
Abstract Herein we establish a relation between quantum irreversibility and the chaotic semi-classical solutions for a spin-boson Hamiltonian system. We obtain quantum averages by numerically integrating the appropriate Liouville-Von Neumann equations of motion and find these averages to be less erratic than the corresponding chaotic semi-classical trajectories. However, the quantum averages are shown to be dissipative as measured by the entropy of the spin subsystem and to suppress the phenomenon of "revivals".
The evolution of classically chaotic quantum systems is analyzed within the formalism of Quantum Pseudo-Probability Distributions. Due to the deep connections that a quantum system shows with its classical correspondent in this representation, the Pseudo-Probability formalism appears to be a useful method of investigation in the field of "Quantum Chaos." In the first part of the thesis we generalize this formalism to quantum systems containing spin operators. It is shown that a classical-like equation of motion for the pseudo-probability distribution ρw can be constructed, dρw/dt = (L_CL + L_QGD)ρw, which is rigorously equivalent to the quantum von Neumann-Liouville equation. The operator L_CL is undistinguishable from the classical operator that generates the semiclassical equations of motion. In the case of the spin-boson system this operator produces semiclassical chaos and is responsible for quantum irreversibility and the fast growth of quantum uncertainty. Carrying out explicit calculations for a spin-boson Hamiltonian the joint action of L_CL and L_QGD is illustrated. It is shown that the latter operator, L_QGD makes the spin system 'remember' its quantum nature, and competes with the irreversibility induced by the former operator. In the second part we test the idea of the enhancement of the quantum uncertainty triggered by the classical chaos by investigating the analogous effect of diffusive excitation in periodically kicked quantum systems. The classical correspondents of these quantum systems exhibit, in the chaotic region, diffusive behavior of the unperturbed energy. For the Quantum Kicked Harmonic Oscillator, in the case of quantum resonances, we provide an exact solution of the quantum evolution. This proves the existence of a deterministic drift in the energy increase over time of the system considered. More generally, this "superdiffusive" excitation of the energy is due to coherent quantum mechanical tunnelling between degenerate tori of the classical phase space. In conclusion we find that some of the quantum effects resulting from this fast increase do not have any classical counterpart, they are mainly tunnelling processes. This seems to be the first observation of an effect of this kind.
Using the Wigner method, it is shown that a classical-like equation of motion for a quasiprobability distribution ${\mathrm{\ensuremath{\rho}}}_{\mathit{W}}$ can be built up, \ensuremath{\partial}${\mathrm{\ensuremath{\rho}}}_{\mathit{W}}$/\ensuremath{\partial}t=(L${\mathrm{^}}_{\mathrm{cl}}$+L${\mathrm{^}}_{\mathrm{QGD}}$)${\mathrm{\ensuremath{\rho}}}_{\mathit{W}}$, which is rigorously equivalent to the quantum von Neumann--Liouville equation. The operator L${\mathrm{^}}_{\mathrm{cl}}$ is equivalent to integrating classical trajectories, which are then averaged over an initial distribution, broadened so as to fulfill the requirements of the quantum uncertainty principle. It is shown that this operator produces semiclassical chaos and is responsible for quantum irreversibility and the fast growth of quantum uncertainty. Carrying out explicit calculations for a spin-boson Hamiltonian, the joint action of L${\mathrm{^}}_{\mathrm{cl}}$ and L${\mathrm{^}}_{\mathrm{QGD}}$ is illustrated. It is shown that the latter operator L${\mathrm{^}}_{\mathrm{QGD}}$ (where QGD stands for quantum generating diffusion), makes the 1/2-spin system ``remember'' its quantum nature, and competes with the irreversibility induced by the former operator. Some ambiguous aspects of ``irreversibility'' and ``growth of quantum fluctuations'' as indicators of semiclassical chaos are discussed.
The nature of classically chaotic quantum dynamics is studied by menas of a numerical time reversal experiment. It turns out that there is a fundamental quantum scale below which the system cannot exhibit classical irreversibility to external perturbations. On the other hand, a paradoxical phenomenon manifesting that quantum irreversibility may exceed its classical counterpart is discovered. These features are explained in terms of semicalssical dynamical theory. Introduction Classically chaotic quantum systems in general can exhibit intrinsinc chaotic behaviour on a quite restricted time scale [1], In particular localization phenomena provide direct evidence exhibiting the suppression of quantum ergodicity [2]. On the other hand, recent studies reveal that a many-dimensional system can mimic some aspects peculiar to chaotic dynamics quite well on an unexpectedly long time scale [3]. A remarkable feature that distinguishes chaotic motion from integrable motion is the sensitive dependence of dynamics on external perturbations. For classical dynamical systems such a sensitivity can be well defined by measuring how two nearby trajectories in the phase space separate in time. However, there has been proposed no systematic way to examine the sensitivity of quantum dynamics to external perturbation. In quantum dynamics, it is not possible to trace a well defined trajectory in phase space, because both quantum uncertainty and chaotic instability make a localized wavepacket spread suddenly over the phase space [4]. There is, however, a simple way in which we may test the quantum sensitivity quantitively. This is the time reversal experiment described below. In this chapter I describe several remarkable characteristics of quantal instability which have been clarified by the time reversal experiment.
We identify a transition to chaos in the semiclassical spin-boson model that occurs for relatively large boson fields, as one of two periodic orbits becomes unstable. We have studied the quantum dynamics in the vicinity of this transition, as characterized by (i) phase-space trajectories followed by quantum expectation values, (ii) spectra of such trajectories, (iii) subsystem entropies for the spin and boson systems, and (iv) growth of operator variances for the boson system. We find that the transition to chaos in the classical system has no apparent effect on any of these variables, in the spin-1/2 case. This is in disagreement with some claims made in earlier studies of this system. \textcopyright{} 1996 The American Physical Society.
Making use of appropriate quantum-classical correspondence we have examined the differential behavior of regular and chaotic trajectories in terms of quantum decoherence, the evolution of average quantities, and variances and entropy in a driven double-well oscillator in contact with the surroundings. Our numerical analysis has shown that even on a short time scale the decay of the average quantum coherence is multiexponential. While the onset of decoherence is much faster for a regular trajectory, the decay is faster asymptotically for a chaotic trajectory compared to a regular one. The coupling of the system to the environment turns chaotic in a regular evolution. The environment also affects the evolution of quantum variances for a regular trajectory almost from the beginning, while it has an insignificant effect on chaotic evolution up to a time after which, for both trajectories, noise in the quantum variances exhibits remarkable suppression, although fluctuations of the reservoir modes, in general, tend to increase the level of variances. We identify three stages of quantum evolution; a short decoherence regime followed by a Liouville flow, the latter regime being dominated by the classical curvature of the potential. This is the regime at which growth of entropy or quantum variances is exponential, the rate being determined by the classical largest Lyapunov exponent. The last stage is the irreversible flow dominated by diffusion which suppresses noises in the quantum variances. \textcopyright{} 1996 The American Physical Society.
We study the Hamiltonian of a two-level system interacting with a one-mode radiation field by means of the Wigner method and without using the rotating-wave approximation. We show that a phenomenon of collapses and revival, reminiscent of that exhibited by the Jaynes-Cumming model, takes place in the high coupling limit. This process appears as irreversible or virtually reversible, according to whether the semiclassical regime is chaotic or not. Thus we find a new mechanism for dissipation in the quantum domain.